Elasticity/Kinematics example 1

Example 1
Take a unit cube of material. Rotate it 90 degrees in the clockwise direction around the z-axis. Calculate the strains. Discuss your results - their accuracy and the reasons for your conclusions.

Solution
The strains are related to displacements by

\epsilon_{xx} = \frac{\partial u}{\partial x};~ \epsilon_{yy} = \frac{\partial v}{\partial y};~ \epsilon_{zz} = \frac{\partial w}{\partial z};~ \gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x};~ \gamma_{yz} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y};~ \gamma_{zx} = \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} $$

Let us consider rotation about the center of the cube. Since the problem concerns a pure rotation, a cylindrical co-ordinate system is appropriate. This problem also provides us a easy way of trying out Maple. Here are the steps that you can follow to find the strains at a point in the cube.



r := \sqrt{x^{2} + y^{2}} $$

\theta := arctan(\frac{y}{x}) $$

x1 := \frac{\sqrt{x^2 + y^2}}{\sqrt{1 + \frac{y^2}{x^2}}} $$

y1 := \frac{\sqrt{x^2 + y^2}\,y}{x\,\sqrt{1 + \frac{y^2}{x^2}}} $$

x2 := -\frac{\sqrt{x^2 + y^2}\,y}{x\,\sqrt{1 + \frac{y^2}{x^2}}} $$

y2 := \frac{\sqrt{x^2 + y^2}}{\sqrt{1 + \frac {y^2}{x^2}}} $$

u := -\frac{\sqrt{x^2 + y^2}\,y}{x\,\sqrt{1 + \frac {y^2}{x^2}}} -\frac{\sqrt{x^2 + y^2}}{\sqrt{1 + \frac {y^2}{x^2}}} $$

v := \frac{\sqrt{x^2 + y^2}}{\sqrt{1 + \frac {y^2}{x^2} }} - \frac{\sqrt{x^2 + y^2}\,y}{x\,\sqrt{1 + \frac {y^2}{x^2}}} $$

epsx := -\frac{\sqrt{x^2 + y^2}}{x\,\sqrt{\frac{x^2 + y^2}{x^2}}} $$

epsy := -\frac{\sqrt{x^2 + y^2}}{x\,\sqrt{\frac{x^2 + y^2}{x^2}}} $$

gamxy := 0 $$ From the above Maple calculation, and noting that there is no motion in the $$z$$ direction, the strains in the cube are

\epsilon_{xx} = -1;~\epsilon_{yy} = -1; \epsilon_{zz} = 0; \gamma_{xy} = 0; \gamma_{yz} = 0; \gamma_{zx} = 0 $$ A pure rigid body rotation should not result in any non-zero strains.

Therefore, the measure of strain we have used is not appropriate for large rigid body motions.